The panel declined to bring charges against Officer Daniel Pantaleo for putting Eric Garner in a chokehold that led to his death.
The prosecutor opposes the move, saying it will hamper witness co-operation.
Similar records were released for a Missouri grand jury investigating the death of unarmed black teenager Michael Brown.
St Louis County Prosecutor Robert McCulloch released the records, with witness names redacted, after the jury declined to charge Officer Darren Wilson for the killing of Michael Brown, 18, in Ferguson.
Michael Brown and Eric Garner’s death sparked protests across the US about police killings and police relations with black and other minority communities.
Eric Garner, 43, was stopped by police for allegedly selling untaxed cigarettes in Staten Island in August and placed in a chokehold by Mr Pantaleo.
In a witness video, Eric Garner, who had asthma, is heard saying “I can’t breathe”. A city medical officer later ruled the death a homicide stemming from the effects of the chokehold.
The US justice department has launched a civil rights investigation in the case.
The New York Civil Liberties Union and other groups who are suing cited the outcry over the Garner grand jury’s decision not to indict despite the video of the incident, as a compelling exception to the normal secrecy of the grand jury.
Disclosure is needed “to restore public confidence in our criminal justice system and to inform the current debate that has begun regarding the role of the grand jury as an instrument of justice or injustice” NYCLU argues in court documents.
The office of Richmond County District Attorney Daniel Donovan said grand jury witnesses in the case came forward to testify “with full assurances of secrecy”.
Making the records public, they argued, would bring an “inevitable result of harassment or retaliation” and a “chilling effect on the very type of witness cooperation that is most desired and the most difficult to obtain”.
Judge William Garnett will hear arguments in the case on Thursday.
The NYCLU cites the decision of Missouri prosecutors to release panel detail in the Wilson case as precedent, but Daniel Donovan says that was also the wrong decision.
News outlets, Daniel Donovan argued, compromised witness anonymity in Missouri by their reporting of the grand jury proceedings and the same could happen in New York.
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